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V.N.M. paper.y and and and M.X., V.H. A.B.S., Barkley, V.H., tools; S. research; A.B.S., reagents/analytic V.H., designed research; V.N.M. performed and Brandt J.A., S. A.B.S., V.H., contributions: Author owo orsodnemyb drse.Eal [email protected] Email: addressed. be may correspondence whom To tpsil oegne tutrlclrfrmn different many for color structural engineer makes a applications. approach to make This possible to constraints. required it specific under parameters opti- color use the target then determine We to parameters. design mization experimental the given this predicts quantitatively for solve that color model To a develop materials. we displays parame- problem, component or tunable cosmetics specific many as require so such color applications are given Furthermore, there a ters. because with applica- challenging nanostructure many a is for making useful However, them making tions. tuned, and broadly fading be to resist colors can similar structural angle, dyes, viewing Unlike of materials. dyed independent scattered are struc- that light have colors nanostructures of tural Disordered interference nanostructure. a from from comes color Structural Significance aigti praharaiyrqie a odesign to way a requires reality a approach this Making NSlicense.y PNAS a igXiao Ming , https://doi.org/10.1073/pnas.2015551118 b eateto hsc,Harvard Physics, of Department a onaAizenberg Joanna , . y https://www.pnas.org/lookup/suppl/ a,c , | f10 of 1

ENGINEERING A B G

C D

H

E

I F

Fig. 1. Overview of design approach. (A and B) Photograph (A) and scanning electron micrograph (SEM) (B) of features from a male Abyssinian roller (Specimen MCZ:Orn:63369. Coracias abyssinica. Africa: Sudan: Blue Nile. El Garef. John C. Phillips; image credit: Museum of Comparative Zoology, Harvard University, C President and Fellows of Harvard College). (C) Photographs of disordered packings of polystyrene particles, showing the structural colors that arise. The radii of the particles increase from left to right. (D) SEM of a disordered packing of 138-nm-radius polystyrene particles. (E) Schematic of the geometry used in our multiple-scattering model. The model is parameterized in terms of experimentally measurable quantities: the volume fraction, complex index of refraction, and radius of the spheres; the complex index of the matrix they are embedded in; the thickness of the film; and the index of refraction of the medium that lies between the viewer and the sample. (F) We calculate the reflectance spectrum by simulating thousands of photon trajectories, a few of which are shown schematically here. (G) The model can predict reflectance spectra that quantitatively agree with experimental measurements, as shown in the plot at Center. Gray area is the uncertainty in the measurement. At Right is a photograph of the measured sample. (H) The model can, therefore, be used to determine the design space—or all of the possible colors—for specific constraints, such as a given type of particle and matrix material. Shown are examples of the colors that can be obtained for fixed material parameters and variable structural parameters. (I) Then, given a target color that is inside the design space, we use optimization to determine the experimental parameters needed to make that color, subject to constraints of our choice, as shown in this schematic.

applications, the nanostructure might be too difficult or expen- matrix phase, and we show that it can make predictions that sive to fabricate. are in quantitative agreement with experiment. We then use the Here, we present a way to solve a more common and chal- model to determine the design space, or all of the possible col- lenging design problem: Given the materials and a simple, ors that can be made, given the experimental constraints. We easy-to-make nanostructure—spherical inclusions in a matrix— demonstrate the design of target colors in two ways: In the first, determine what colors can be made and what structural parame- we choose target colors from the design space for specific mate- ters (particle size and volume fraction, for example) are required rial systems. In the second, we target a given point in a perceptual to make a given color. Compared to the problem of determining colorspace and use optimization to determine the experimental the materials and structure required to make a given color, our parameters that produce this color. Overall, the approach that problem is complicated by the potential absence of solutions that we demonstrate (illustrated in Fig. 1 E–I) gives us the freedom meet the needs of the application. Determining whether a solu- to precisely design and control angle-independent colors under tion exists requires a way to predict all possible colors for a given constraints of our choice, which opens possibilities that go well set of constraints. beyond biomimicry. Tackling this design problem requires an accurate model for how light interacts with disordered composite materials. Such a Model model must account for both absorption and multiple scattering Approach. Light propagating through a composite material is of light, which affects the color saturation. Because structurally scattered when the refractive indices of the component mate- colored samples are composite materials, each component has rials differ and is absorbed when either of the materials has a its own wavelength-dependent index of refraction and absorption refractive index with a nonzero imaginary component. The scat- coefficient. Even if the components are “transparent” dielectrics, tered waves can then interfere with one another. Furthermore, a small amount of absorption can change the reflectance spec- depending on the refractive indices and nanostructure, light trum. As we shall show, these effects must be modeled carefully might scatter repeatedly before exiting the material. Thus, mod- to quantitatively predict color. eling structural color requires knowing the complex refractive In what follows, we describe a Monte Carlo model of light indices of the materials, the nanostructure, and the detection transport in disordered packings of spherical inclusions in a geometry.

2 of 10 | PNAS Hwang et al. https://doi.org/10.1073/pnas.2015551118 Designing angle-independent structural colors using Monte Carlo simulations of multiple scattering Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 lhuhfieruheshsalre fetta oreruhes ems con o oht rn h oe noqatttv gemn with agreement scattering quantitative have multiple of into systems simulations Appendix model Carlo most Monte ( SI the using therefore, colors measurement bring structural sample; the to angle-independent the Designing of both hit Coarse uncertainty for al. et cannot one. the account Hwang light show must and regions we that zero Gray roughness, means (28) between air. course surface slope is in than the large and spheres effect of a film polystyrene larger slope bound, the 218-nm a rms upper of on has experiment. dimensionless no film roughness incidence the (F 85-µm is fine is upon 1.1. an Although there parameter particle and of roughness While is single 0 measurements coarse surface. line a between mental The flat dashed of roughness surface. including a The scale coarse smooth, model, for fields. the a locally the zero scattered though on from is the tilted, roughness calculated 85- and a of encounters spectra Appendix an to amplitude Reflectance (SI that of trans- corresponds the (D) measurement light measurements roughness to represent cross-section. the of different film. experimental opposed scattering curves of fraction to at the as differential the uncertainty compared the trajectories wavelength, exits the and the absorption, each of packet integrate show field, of at thousands the we contribution regions incident reflected simulating where until the the are After particle, excluding repeats depict that new the rules. and process arrows packets a of above form This The of into surface the the wavelength. fraction sample. scatters the using obeying the packet the the by simulation the counting through on calculated illustrated Then, by Carlo (C are travel depend as index. spectrum absorbed. sampled function Monte both step, refractive phase reflectance is or a this effective and and the size mitted for during complex distribution medium, step calculate absorbed trajectories the step-size effective the we be by photon the where wavelengths, complex can determined Both of sample, the packet as function. Rendering the the and phase in into (B) of arrow the factors step Part orange from structure a length. the sampled takes and scattering is of first which the width packet direction, the is Each propagation in mean medium. decrease whose effective the distribution, an by probability in exponential propagating an and from scattering “packets” photon 2. Fig. to step- used a been a have from methods from sampled Carlo Monte sampled directions 23). (22, into are function scattering phase that and steps propagate distribution taking “packets” size system, photon a approach, this through In trajectories. ton color they the but controls which peak, scattering, reflection (20). multiple saturation the mod- for of These account wavelength not 21). do the 20, predict based can (2, els models in approximations parameterized single-scattering effective-medium nat- not on include more are a parameterization with they Approaches ural because properties. design experimental dif- of in and terms use intensive computationally to are ficult but effects, do such (13–19) techniques capture finite-element finite-difference- and as (FDTD) such time-domain methods in Numerical not, of materials. does structurally colored mixtures of theory characteristic in effects transfer interference radiative colors capture general, However, predicting 12). for (11, transfer extensively which paints (10), radiative used theory been is Kubelka–Munk the has venerable particular, most in and, The theory color. and scattering 0 z A C oegnrlapoc sMneCrosmlto fpho- of simulation Carlo Monte is approach general more A between relation the modeling to approaches many are There incident absorbing matrix 180° fields 225° 135° ato fMneCromto.W iuaetetaetre of trajectories the simulate We method. Carlo Monte of Cartoon (A) color. structural angle-independent for model Carlo Monte of Overview absorbing 270° 90° particle 315° 45° fields decay 0° ceai hwn o bopini mlmne ntemdl na bobn ytm h nietfilsdcya they as decay fields incident the system, absorbing an In model. the in implemented is absorption how showing Schematic ) scattered 2. Scatter 3. Repeat 3 D

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https://doi.org/10.1073/pnas.2015551118 y 0 400 experiment roughness with no 500 roughness with coarse Wavelen 600 g fine roughness with coarse and th (nm) PNAS 700 | f10 of 3 x 800 ).

ENGINEERING Technique. Our model accounts for constructive interference In the validation experiments that follow, we do not fit through the phase function. There are two contributions to this each measurement individually; instead, because we expect the function: the form factor, which describes the angle dependence roughness values to largely depend on the sample-assembly of scattering from individual particles and can be calculated from method, we fit the values to all samples fabricated with the same Mie theory; and the structure factor, which describes the con- technique. structive interference of waves scattered by different particles and can be calculated from the Percus–Yevick equation (2, 29). Results This theory allows us to compute the structure factor analytically Model Validation. We validated the model by comparing the as a function of volume fraction, assuming that the structure is predicted and measured reflectance spectra for samples with equivalent to that of a hard-sphere liquid. It does not require different physical parameters. In each of the simulations, we us to know the positions of the particles. Although the structure took the average of 20,000 trajectories at each wavelength. For factor accounts for constructive interference within each trajec- such a large number of trajectories, the Monte Carlo uncer- tory, we do not model constructive interference among different tainty in the predicted reflection spectrum is much smaller (SD trajectories. 0.4%) than the uncertainty of the experimental spectrum, which We assume that the scattering occurs in an effective medium is determined by taking measurements from different parts of determined by the average properties of the material (Fig. 2A). the same sample. Each measurement was done by collecting Our effective-medium theory, which is described in more detail all of the light scattered into the reflection hemisphere (see SI in SI Appendix, is based on the Bruggeman approximation (30), Appendix for a full description of the sample fabrication and which can account for complex refractive indices, though not for characterization). near-field effects (Discussion). We first examined the effect of the particle radius. We cal- To account for wavelength-dependent absorption, we use a culated reflectance spectra for packings of polystyrene parti- modification of Mie theory that accounts for absorption in both cles in air for three different polystyrene radii: 94, 109, and the particles and matrix. In an absorbing system, the scattered 138 nm. As shown in Fig. 3A, the model accurately captured fields are absorbed as they propagate away from the scatterer, the redshift of the reflectance peak with increasing particle such that the differential scattering cross-section of a particle size, while also reproducing a rise in scattering toward small decreases with distance (31–33). This consideration applies not wavelengths. The predicted spectra quantitatively matched the only to systems with absorption in the matrix, but also to those data in both the location of the reflectance peak and the with only absorbing particles, because in both cases, the imagi- reflectance magnitude across the visible range with only small nary index (and, hence, the absorption coefficient) of the effective deviations. The model also captured the broadening and aver- medium is nonzero. Therefore, when the particle or matrix has aging of the peak when two particle radii were mixed, which a complex refractive index, we obtain the total scattering cross- validates our implementation of polydispersity (SI Appendix, section by integrating the differential scattering cross-section at Fig. S1A). As a result, the colors predicted by the model visu- the surface of the scatterer (34, 35). We then account for absorp- ally matched the color renderings calculated from the measured tion of the photon packets traveling through the effective medium reflectance. with an exponential decay function based on the Beer–Lambert Having shown previously that a small amount of absorption law. Lastly, we correct for the variation in the amplitude of the can alter the reflection spectrum (Fig. 2D), we must now fur- incident field as a function of position on the sphere (34) (Fig. 2C). ther confirm that our model accurately captures the effects For more details on the model, see SI Appendix. of absorption in experimental samples. We made concentrated Modeling absorption leads to better agreement between the samples of polystyrene spheres in water, and we tuned the predicted and measured reflection spectrum (Fig. 2D). The amount of absorption by adding varying amounts of carbon small amount of absorption in polystyrene particles, for exam- black. To model these samples, we assumed a matrix with a ple, changes the predicted reflectance spectrum from that of a real refractive index of water and an imaginary index corre- sample without absorption, especially at short wavelengths. sponding to the concentration of carbon black (SI Appendix). We also account for surface scattering, which can arise from Thus, we neglected any scattering from the carbon black parti- the roughness inherent to most experimental samples. We model cles, which is a reasonable approximation, given that the carbon this roughness at two different scales: coarse and fine. Coarse black particles are approximately 10 nm, much smaller than the roughness is large compared to the wavelength, such that inci- wavelength. We again found that the model accurately predicted dent light encounters a locally smooth surface that is angled the reflectance and color of samples with varying amounts of with respect to the incident direction. We model coarse rough- absorption (Fig. 3B). ness by accounting for the refraction of light when it encounters In addition, we explored the validity of the model over a range the boundary of the film (Fig. 2E). Fine roughness arises from of film thicknesses. In the thickest sample, the thickness was wavelength-scale features, such as particles protruding from the much larger than its transport length, which is the length scale surface. To model fine roughness, we sample the initial step at which the direction of light is randomized. In the thinnest size of a trajectory from the scattering properties of a single sample, the thickness was smaller than its transport length at all nanoparticle, ignoring the contribution of the structure factor. wavelengths. The model agreed with experiment when the thick- For many of the samples we examine, such as those dispersed ness was large, but started to deviate from experimental data in in a liquid, we cannot easily measure the roughness parameters. thin samples and at large wavelengths (Fig. 3C). The discrep- Indeed, as we note in Discussion, the roughness parameters can ancy likely arises because, for very thin samples, the distinction be viewed more generally as correcting for the failure of the we make in our model between surface scattering and bulk scat- effective-medium approximation at the boundary of the sam- tering starts to break down. However, most structurally colored ple. Therefore, we determine these parameters by fitting them samples are not as thin as the thinnest sample we show here. to measurements. When we do this, we find that including coarse Furthermore, the predicted colors in all samples were similar to and fine roughness brings the model into quantitative agreement those of the experimental samples, despite the deviations in the with experiment (Fig. 2F). Although the parameters are fitted, predicted reflection spectrum for thin samples. they are constant with wavelength. Therefore, the agreement In SI Appendix, we further validate the model on bidisperse between the fitted model and the data as a function of wave- samples (SI Appendix, Fig. S1A) and samples with varying volume length shows that our model for roughness captures a physical fraction (SI Appendix, Fig. S1B). For the volume-fraction exper- effect of the sample boundary. iments, we prepared samples of polystyrene spheres in water, in

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ENGINEERING (or perceived vividness), and luminance. Each of these can be A calculated by transforming the computed reflectance spectra used to generate Fig. 4A to the CIE 1976 L∗, u∗, v ∗ (CIELUV) perceptual colorspace. We found that small particle radii gave rise to colors in the blue and green, as expected, but red hues remained inaccessible, in agreement with the results of Scher- B tel and coworkers (36, 37). We also found that increasing the volume fraction can significantly increase the chroma and blue- shift the hue, while decreasing the luminance. Increasing the thickness did not affect the hue. Instead, it slightly increased the chroma and luminance at small imaginary indices, but not at the largest imaginary indices, where the absorption length becomes comparable to or smaller than the sample thickness C (20). When we replaced the air matrix with water, we found that increasing the radii led to a range of browns, instead of pinks and purples (Fig. 4B), because the lower index contrast between polystyrene and water led to flatter and broader reflectance peaks. Increasing the volume fraction blue-shifted the hue and increased the chroma. The thickness did not affect the hue or chroma, but only increased the luminance when the absorption was low, as in the polystyrene-in-air system. In both systems, Fig. 5. Designing colors from the gamut. Each plot shows the reflectance increasing absorption only decreased the luminance and did not spectra of the target color (dotted line) and of the color that is achieved change the hue or chroma. (solid line) in a sample made using the model parameters for the target. To demonstrate the predictive power of the model, we made At Right is a colormap showing the CIE chromaticity coordinates of the tar- three colors from these gamuts (outlined swatches in Fig. 4). The get (circles) and achieved (crosses) colors. The target colors are chosen from colors were chosen from across the visible spectrum. We made the color gamuts in Fig. 4. The parameters are as follows. (A) Blue target: a green sample with polystyrene particles in air and a blue and radius 82 nm, volume fraction 0.42, thickness 130 µm, and matrix imaginary index 0.0003i, corresponding to 0.08% by weight of carbon black. (B) Green a light brown sample with polystyrene particles in water. We target: radius 110 nm, volume fraction 0.4975, thickness 40 µm, and matrix made samples with parameters as close as possible to the val- imaginary index 0.0017i, corresponding to a carbon black concentration of ues used in the simulations, and we found that the target and the 0.42% by weight. (C) Brown target: radius 112 nm, volume fraction 0.35, achieved colors agreed well, with some small deviations at large thickness 110 µm, and matrix imaginary index 0.000055i, corresponding to wavelengths (Fig. 5). 0.016% by weight of carbon black. The uncertainties in the achieved spectra are shown in gray and represent two SDs about the mean of measurements Finding the Parameters to Design a Target Color. In addition to tar- from 11 (blue spectrum), 19 (green), and 12 (brown) locations on the sample. geting colors in the gamut, we can also target a particular color in a colorspace. We used the perceptual colorspace defined by the CIE 1976 L∗a∗b∗ (CIELAB) coordinates (38) because appli- When we minimized the difference between the target color cations such as cosmetics or coatings are aimed at the human and the color obtained from the model, we found a good match eye. Using such an approach increases the number of available in CIELAB space (Fig. 6 D and E). Note that we matched the designs because we can exploit the eye’s insensitivity to variations color, and not the reflectance spectrum, because matching the in certain parts of the spectrum. spectrum may not be possible for the given materials. Indeed, To implement this approach, we chose a target color in the spectrum of the best-fit solution had a narrower peak than CIELAB coordinates and then used Bayesian optimization (39) that of the target, with the target having a larger reflectance at to find the model parameters that minimized the sum of squared wavelengths less than 450 nm (Fig. 6F). Because the eye is insen- differences between the target CIELAB coordinates and those sitive to such short wavelengths, the best-fit solution need not corresponding to the modeled reflection spectrum. We call the duplicate this feature to match the color in the CIELAB space. optimal solution the “best fit” to the target. We found that the best-fit solution had a CIE76 color difference We chose the color of the mountain bluebird as our tar- of 3.9 from the target color, which is close to the just-noticeable get (Fig. 6A), because the feathers show an angle-independent difference (JND) of 2.3 (40). structural blue (Fig. 6B). This color arises from a porous inter- We made a film with parameters as close as possible to those nal structure (Fig. 6C) that likely evolved to meet constraints of the best fit. We found that both the resulting spectrum and other than color, including (perhaps) minimizing weight and color were close to those of the best fit, as shown in Fig. 6 D– maximizing insulating ability and water repellency. F. The CIE76 color difference between the achieved and target We imposed a different set of constraints. Because the blue- colors was 5.1, larger than the difference between target and bird’s “inverse” structure of pores inside a solid matrix is not as best-fit, but less than 2.5 times the JND. The difference between easy to fabricate as a “direct” structure of solid spheres in air the best-fit and achieved colors may come from the values of or water, we designed the color using a direct structure instead. the roughness parameters we used in the model. Although we Furthermore, we constrained the materials to those we had on used the same preparation technique and, thus, the same fine- hand: polystyrene spheres and a matrix of either air or water. and coarse-roughness values as for the polystyrene films from We used Bayesian optimization to determine the optimal parti- Fig. 3A, the true roughness values of the sample might dif- cle radius, volume fraction, film thickness, and concentration of fer from these values. Nonetheless, the agreement between the carbon black. To ensure that the optimal values can be exper- achieved and target colors shows that one can design the color imentally achieved, we set ranges for these parameters: The of the feather without mimicking its structure, instead using a particle radius was 74, 101, 110, 112.5, or 138 nm; the thickness system that satisfies a different set of constraints. was between 20 and 150 µm; and the range for the matrix imag- inary index was between 0 and 0.005i. We used the same values Discussion of roughness as in the samples in Fig. 3A: fine roughness of 0.5 Having shown that our model can be used to design colors, and coarse roughness of 0.9. we now examine its limitations and why it works despite these

6 of 10 | PNAS Hwang et al. https://doi.org/10.1073/pnas.2015551118 Designing angle-independent structural colors using Monte Carlo simulations of multiple scattering Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 ersn w D bu h eno esrmnsa he oain ntesml.Nt htw onttyt ac h oe n agtspectra; target and model the match to try not do we that Note sample. the coarse reflectance on and achieved the 0.5, locations for of three regions roughness the at Gray in fine sample. measurements performed black, line) 101-nm-radius of is (dashed carbon follows: optimization polystyrene-in-air mean of the and as the weight line), instead, are by about (dotted constraints 0.9% fit SDs model thickness, the two line), film satisfies (solid represent 50-µm that feather matrix, bird solution air of (B) best-fit spectra an color. the Reflectance in of target (E 0.54 0.9. parameters of the of fraction The for roughness volume colors. a measurement achieved at and reflectance spheres best-fit, of polystyrene target, area the (C denotes of bird. renderings Circle the of Willians). for back credits S. the Image Robert from feather Martinsdale. a Meagher. of Montana: Photograph States: United America: etltasotlnt sa es ortmslre hnthe experi- than the larger because times four range least this at Near-field study. is in wavelength. our length neglected in transport used be mental Appendix, range can (SI the 0.5 covers effects scattering roughly which of in S2), ratio peaks Fig. (42) radius-to-wavelength al. the a ratio et that to Aubry the up by find of measurements We experimental function match wavelength. a strength to transport as index. strength radius the effective scattering of Bruggeman to the the calculate wavelength and the We factors both the using structure by of and calculated is form scatter- ratio length transport The the the where air. length, is the in calculate strength particles we polystyrene ing corrections, of near-field strength of scattering absence the despite its and absorption handle wavelength. directly on refrac- can dependence complex it for Therefore, account can indices. it Our tive near-field that for advantage thickness. account the not but sample does effects, it that the disadvantage approximation, the of medium has which effective cutoff Bruggeman the a uses approach as dielectric absorption. absorption approx- by real for limitation imating this for account for compensate only not coworkers and valid does Schertel is therefore, ECPA and, scat- multiple the permittivities of Second, amount the (20). minimize tering to samples, chosen colored is structurally thickness many the in whereas before sample, times the many exits diffusion scatters it the light when First, only reasons. valid two is for approximation purposes our for suitable not structural predict in to ECPA approximation the diffusion used colors. (36) the of al. with packing et dense concert Schertel a (ECPA), (41). in scatterers approximation effects spherical near-field potential for theory, coherent corrections effective-index includes alternative energy An the spheres. and index called the effective the of the determines index between calcu- that difference the the the the difference is index of using function the phase all calculate example, underlies we For (30), which lations. average index, weighted effective Bruggeman This refrac- effective index. an with prop- effective tive medium light the homogeneous events, a of scattering through between agates that that is assume We approximation medium: central The limitations. 6. Fig. einn nl-needn tutrlclr sn ot al iuain fmlil scattering multiple of simulations Carlo Monte using colors structural angle-independent Designing al. et Hwang D ABC oudrtn h u prahcretypeit spectra predicts correctly approach our why understand To is it but spectra, reflectance predict can ours, like model, Their htgaho aemuti leid(pcmnMCZ:Orn:190556. (Specimen bluebird mountain male of Photograph (A) colorspace. the in color specific a Targeting A–C I hoaiiyclrmpcmaigtetre crl) etfi cos,adahee oos(rage netit hw nga) (F gray). in shown uncertainty (triangle; colors achieved and (cross), best-fit (circle), target the comparing map color chromaticity CIE ) uemo oprtv olg,HradUniversity, Harvard Zoology, Comparative of Museum : L * , EF a * , b * E facosscino h ete’ nenlsrcue bandatrfcsdinba milling. beam focused-ion after obtained structure, internal feather’s the of cross-section a of SEM ) space. ogns aaeesms efitdt xeietldata. experimental to fitted be must parameters roughness for values appropriate the parameters. give these as not may and such microscopy fine- techniques force atomic The measurement sample. topographical effects, the therefore, these is of and, of all which boundary for factor, compensate the structure parameters coarse-roughness at the of defined approxima- effects well material effective-medium the not includes homogeneous the use a we Furthermore, and tion other. side the one surface the on on same at the particles particles in other whereas the embedded sides, have particle with all a on light for medium does of derived (effective) occurs theory is interaction theory breakdown Mie initial Mie because The the film. but describe sample. topography, accurately the of approxi- not of because effective-medium only surface the not the of parameters at topography breakdown roughness mation the the the for generally, roughness for more account the account But, they samples. introduced the that we of argued When we between agreement experiment. parameters, quantitative and achieve model to roughness surface the of spectrum the shown. the when have we to even as matched colors, color, target be perceptual cannot design spectrum to reflection used our be Therefore, spectra. can reflectance model full on three than on rather based at receptors, colors system—especially detect eyes visual our human the why wavelengths—because of the long reason advantage of take capacity fundamental the we color the Furthermore, to limited well. is comparable so works This be model material. length the the transport of the scat- and thickness multiple weak that be requires tering saturation Color materials. colored important. trajectories, become might for, photon account not scattered does model our multiply which between multiple stronger, is weak scattering interference multiple magni- of When of length. order regime transport similar the a the as on sug- tude in is 3C) thickness best film (Fig. the works where thicknesses scattering, model large our and that small gest Furthermore, at the wavelength. when discrepancies the or to strengths, the scattering comparable high is at length work transport to it expect not do

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ENGINEERING Indeed, as we have shown, the parameters need not be fitted AB C to measurements for each individual sample; instead, one can use the same values for all samples that are made with the same assembly technique. To improve the predictive accuracy, one can use an iterative design approach: First, make an initial guess for the roughness and find the model parameters that best fit a target color; second, make the sample by using the best-fit parameters and fit the model to the data to improve the estimates of the roughness parameters; and, third, use the improved estimates to find parameters that give a better fit of the model to the target D E (Fig. 1D). The power of our model lies in providing a physical under- standing of how the experimental parameters change the color. This insight enables a rational design approach for the nano- structure. Consider a case when the materials are prescribed— for example, polystyrene in air or water—but the structure can be varied—for example, by making composite particles. This sit- uation arises in many applications: The constituent materials must meet certain requirements (regulatory or other), but the F G spatial arrangement of these materials may be unconstrained. Because there are infinite possible arrangements that differ from solid spherical particles in a matrix, finding the optimal arrangement for a target color is a very difficult design prob- lem. We can, however, use the physical intuition provided by the model to choose a nanostructure that produces a particular color. As an example, we consider making colors that are outside the gamut of a system of polystyrene particles in air, yet use the same materials. Solid polystyrene spheres in air tend to have low Fig. 7. The model allows us to explore the limits of colors that can be saturation or chroma, particularly in the red, as shown in the achieved with different configurations of a set of materials. All plots assume gamut of Fig. 7A. The low saturation comes from scattering at an imaginary refractive index for polystyrene of 2 × 10−5i, volume fraction short wavelengths, as shown in the purple spectrum in Fig. 7B. of 0.64, thickness of 20 µm, coarse roughness of 0.9, and fine roughness The short-wavelength scattering comes from the large scattering of 0.5. (A) Color gamut for polystyrene (PS) particles in a matrix of air as a function of particle radius. (B) Reflectance spectra for colors denoted by cross-section of polystyrene particles in the blue (20). The model the black arrows in A. (C) Transport length as a function of wavelength for shows that this large cross-section gives rise to multiple scat- the purple system in B and the pink system in D. (D) Reflectance spectra tering. The propensity for multiple scattering can be described for three colors chosen from the gamut for particles with air cores and PS by the transport length, which is small at short wavelengths shells in a matrix of water. (E) Color gamut for the core–shell system as a (Fig. 7C). function of core and shell radii. Black rectangles denote the samples whose To decrease this scattering, we designed an alternative reflectance spectra are shown in D. (F) Reflectance spectra for three colors arrangement of the materials. First, we inverted the particles from the gamut of a core–shell system with absorption added to the matrix. into air cores with polystyrene shells (2) to reduce the scattering (G) Color gamut for core–shell system as a function of core radius, shell cross-section in the blue. Second, we placed the core–shell parti- radius, and matrix imaginary index. Black rectangles denote the samples whose reflectance spectra are shown in F. cles in a matrix of water to decrease the index contrast between the shell and the matrix (Fig. 7D). Because the resulting colors are still desaturated (Fig. 7E), we suppressed multiple scattering by adding an absorber to the water (Fig. 7F). We used the model only certain materials can be used. The predictive power is also to determine what absorber concentrations lead to optimal sat- important for applications such as paints and coatings, where the uration. The resulting gamut showed colors and saturations that color might change substantially as the suspending liquid dries are different from those of polystyrene particles in air—in par- and the refractive-index contrast increases. ticular, we now see orange and brown hues that arise due to the Compared to FDTD and similar methods, our model decreased scattering at short wavelengths (Fig. 7G). addresses a smaller range of nanostructures—those involving From this example, we see that loosening the restrictions on disordered arrangements of spheres—but it is much faster than the arrangement of the materials increases the size of the design FDTD because particle positions need not be specified. Also, space, but also makes it possible to access new colors. The phys- no experimental measurements or simulations of particle posi- ical intuition provided by the model is critical for exploring this tions are needed to calculate the spectrum. One disadvantage larger design space. of our method is that it is not as accurate as FDTD, because it makes approximations such as the effective-medium approxima- Conclusion tion and neglecting near-field coupling. But, as we have shown, Engineering materials with prescribed structural colors requires our model is in quantitative agreement with experiment, despite a way to predict the color from the nanostructure. Doing so effi- these approximations. Furthermore, unlike FDTD, our model ciently requires accurate predictions, so as to minimize iteration is parameterized in terms of experimental quantities, and the between experiment and simulation. We have demonstrated a results can be interpreted in terms of collective and resonant model of multiple scattering in disordered packings of spheres scattering effects. The parameterization facilitates the use of that produces accurate predictions. The model is parameterized optimization methods to design colors, which is difficult with in terms of experimental quantities, such as the volume frac- FDTD because the entire nanostructure must be specified at tion and material optical properties. As such, it can be used each optimization step. The interpretability of the results can be to design structurally colored materials that meet specific con- used to rationally design variant nanostructures with wider color straints, making it particularly useful for applications in which gamuts, as we have shown.

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J. 1. ial,temdlcnb sdt rdc h nl dependence angle the predict to used be can model the Finally, and geometries complex more to extended be can model Our multi- weak of regime the in model the validated have We to restriction current model’s the that however, note, We particles nonspherical to model our extend to possible is It h cre macaw. scarlet the 2014). 154. vol. 2010), NY, York, ls-akn htncstructures. photonic close-packing Physik fabrication. and modeling, colors. neutral create to diameters different with Langmuir particles melanin-like Mixing ing: oration. granules. melanin mimic that particles C Chem. black Mater. polydopamine from materials color nanoparticles. melanin nanoparticles. melanin palette. color (2017). integumentary the on effects their beetles. certain and feathers, bird (2014). glasses, photonic in color Mater. mrhu imn-tutrdpooi rsa ntefahrbrsof barbs feather the in crystal photonic diamond-structured Amorphous al., et 9–0 (1931). 593–601 12, 9924 (2010). 2939–2944 22, tml-epniesrcual ooe lsfo iisie synthetic bioinspired from films colored structurally Stimuli-responsive al., et Biomacromolecules i-nprdsrcua oosproduced colors structural Bio-inspired al., et nprto rmbtefl n ohwn cls Characterization, scales: wing moth and butterfly from Inspiration al., et 8433 (2017). 3824–3830 33, pigr New Springer, Sciences, Optical in Series (Springer Physics Color Industrial immtciorpcnnsrcue o tutrlcoloration. structural for nanostructures isotropic Biomimetic al., et plcto fLgtSatrn oCoatings to Scattering Light of Application 2–2 (2015). 720–724 3, rc al cd c.U.S.A. Sci. Acad. Natl. Proc. hm Mater. Chem. Nano ACS rg ae.Sci. Mater. Prog. 6–6 (2015). 660–666 16, 4456 (2015). 5454–5460 9, p.Express Opt. 5652 (2016). 5516–5521 28, 79 (2015). 67–96 68, 09–00 (2012). 10798–10801 109, 43–43 (2010). 14430–14438 18, hl rn.Bo.Sci. Biol. Trans. Phil. via efasml fsynthetic of self-assembly Srne,NwYr,NY, York, New (Springer, hs Rev. Phys. etcrf Technische Zeitschrift 20160536 372, 062302 90, Adv. J. 5 .P hwn .DAesnr,X u pia mgn oaiisfrbiomedical for modalities imaging Optical Fu, X. D’Alessandro, B. Dhawan, light-based P. on A. curvature surface 25. of Influence Tan, Z. Wei, W. Chen, J. Shi, S. Ding, C. 24. 8 .vnGnee,M tvii .J oneik ifs n pclrrflcac from reflectance specular and Diffuse Koenderink, J. J. Stavridi, M. Ginneken, van tissues. B. in transport light 28. of modeling Carlo Monte of methods Review Carlo Liu, Q. Monte Zhu, Obara, C. T. Thomassin, 27. M. Tindel, S. Lacaux, C. Vinckenbosch, L. 26. trans- light of modeling S. Carlo N. MCML—Monte 23. Zheng, L. Jacques, L. S. Wang, L. 22. Effects Manoharan, N. Maiwald V. L. Park, 21. G. J. algae. marine Magkiriadou, in S. color Stephenson, Structural B. Vignolini, A. S. Hwang, Brodie, V. J. 20. Wilts, D. B. Chandler, J. C. 19. plasmonic Xiao on M. based 18. printing color Structural Galinski Yang, X. H. Luk, 17. S. T. Gao, J. the Cheng, in F. mechanism color 16. Structural Lee, C. C. Lo, L. M. 15. eerhFlosi rn G-145.I a efre npr tthe at 1541959. Grant part NSF in by supported performed Graduate Systems, was Nanoscale NSF for It by Center DGE-1144152. Harvard and Grant DMR-2011754; Fellowship Grant BASF and Research NSF the Science for and under Research Materials Corporation Center Eldridge Harvard BASF Engineering the by Kate by Alliance; supported Har- Research and Zoology, is Northeast Comparative work Trimble of This from Museum University. borrowed Jeremiah the vard were of which and Department specimen, Ornithology and work; the feathers bird this the with nanopar- in assistance polystyrene the used providing for GitHub ticles Park Jin-Gyu Konradi at discussions; Rupert helpful and Vacano, for software von Bernhard Schroeder, Mark source Franklin, Melissa open and ACKNOWLEDGMENTS. free as available of (48). is (https://github.com/manoharan-lab/structural-color) code source model The the (47). (https://doi.org/10.7910/DVN/2Q2JRU) Dataverse Availability. Data implemented. are scattering particles, surface multilayered and and effects, trajectories polydisperse boundary how absorption, on how details and including calculated model, are in Carlo given Monte polydisper- the are of radius, indices, description refractive particle and as and thickness, Appendix, such film spectra, samples fraction, the reflectance volume the including sity, measuring of methods, properties samples, experimental measuring fabricating and for materials procedures the of Descriptions Methods and Materials structurally them—in chroma, between materials. tradeoffs dependence, colored the angle as determine well of inci- hue—as help limits and the could physical of approach fundamental function This the a angles. can as detected one and spectrum However, dent reflection light. ambient the in simulate simulating look also thus would hemisphere, sample the reflection trajec- how the all counting into by color scattered reflectance of tories detector. the variation calculate and weak we source above, a the shown is between angle there the reality, with in color, the describe * esrmnsi gr 0o e.2) h ILBcolor CIELAB the 20), ref. of 10 figure in measurements 1G; (Fig. sample typical a For ifrnebetween difference hsdfeec sntcal yee u smc ekrta hto an of that than weaker much is but eye, by noticeable is difference This 6D. Fig. in n o ntedrcin of direction, the on not and ree tutr,prl eas h tutr atrdpnsol ntemagnitude, the on only depends factor structure the because partly structure, ordered h N—oprbet h ifrnebtentetre n civdsamples achieved and target the between difference the to JND—comparable the Optic. applications. fruit. (2016). stone of measurement nondestructive tissue. in migration (2006). photon of eling metamaterials. og surfaces. rough tissues. biological in propagation light for tissues. multi-layered (1995). in port (2018). 11352–11365 colloidal disordered in color structural angle-independent materials. on scattering multiple of Materials Optical Advanced (2017). e1701151 3, absorption. light perfect of metasurfaces Optic. oe,A ibr,R aiwk,Otmzto fteMneCrocd o mod- for code Carlo Monte the of Optimization Maniewski, R. Liebert, A. Zołek, ˙ 592(2013). 050902 18, (2014). A399–A404 53, tal. et aeil n Methods. and Materials hs Rev. Phys. wl peecntuto o tutrlcolors. structural for construction sphere Ewald al., et clbe lr-eitn tutrlclr ae nnetwork on based colors structural ultra-resistant Scalable, al., et iisie rgtnnrdsetpooi eai supraballs. melanin photonic noniridescent bright Bioinspired , EERv imd Eng. Biomed. Rev. IEEE ih c.Appl. Sci. Light pl Optic. Appl. 16 l h esrdsetaaeaalbea h Harvard the at available are spectra measured the All ◦ 164(2020). 012614 101, and etakRp aj,KihTs,Jrm Fung, Jerome Task, Keith Darji, Rupa thank We q. 76 604 (2017). 1600646 5, 3–3 (1998). 130–139 37, ◦ 123(2017). e16233 6, wt epc otenra)i .,o bu . times 2.5 about or 5.6, is normal) the to respect (with opt ehd rg.Biomed. Progr. Methods Comput. 99 (2010). 69–92 3, opt ehd rg.Biomed. Progr. Methods Comput. IAppendix SI https://doi.org/10.1073/pnas.2015551118 c.Rep. Sci. ah Biosci. Math. opt lcrn Agric. Electron. Comput. 14 (2015). 11045 5, aii blumei Papilio loicue technical a includes also 86 (2015). 48–60 269, ∗ nteresults the In PNAS pi Express Optic butterfly. 200–206 121, 131–146 47, | .Biomed. J. 50–57 84, c.Adv. Sci. f10 of 9 Appl. 26, SI

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10 of 10 | PNAS Hwang et al. https://doi.org/10.1073/pnas.2015551118 Designing angle-independent structural colors using Monte Carlo simulations of multiple scattering Downloaded by guest on September 26, 2021